Finite dimensional integrable structure in systems with infinite degree of freedom

无限自由度系统中的有限维可积结构

基本信息

批准号：
10640165
负责人：
TAKASAKI Kanehisa
金额：
$ 2.11万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

This research project aimed to search for various finite dimensional integrable systems in systems with infinite degrees of freedom, and to elucidate their mathematical structures. Related issues on geometry and non-integrable systems were also investigated.The head investigator obtained interesting results on Whitham deformation equations, isomonodromic deformations, systems arising in supersymmetric/topological gauge theories and Calogero-Moser systems, all of which are mutually related. Firsly, he could find an explicit form of the Whitham deformation equations for asymptoci description of isomonodromic systems on the Rie-mann sphere. Secondly, an extension, to a torus, of such isomonodromic systems was achieved on the basis of methods in solvable spin sysmtes and conformal field theories. Thirdly, he clarified the roles that the tau functions and the Whitham deformations play in four dimensional supersymmetric and topolotical gauge theories. Finally, he considered the elliptic Calogero-Moser systems, which are also closely related to four dimensional supersymmetric gauge theories, and discoveref that a non-autonomous analogues of those systems give an example of isomonodromic dermations on a torus.The achievement due to other members of the project group respectively ranges over low-dimensional topology, stochastic analysis of hypoelliptic operators, complex dynamical systems, transcendental number theory related to nonlinear dynamical systems, celular automata, intelligent agents, hypoelliptic and hyperbolic equations with infnite degeneracy,

该研究项目旨在在具有无限自由度的系统中搜索各种有限的尺寸整合系统，并阐明其数学结构。还研究了有关几何学和不可融合系统的相关问题。主管研究员在Whitham变形方程，异构粒细胞变形，在超对称/拓扑理论中产生的系统和Calogero-Moser系统中获得了有趣的结果，所有这些系统都是相互关联的。 Firsly，他可以找到对Rie-Mann球体上异构粒细胞系统的渐近描述Whitham变形方程的明确形式。其次，基于可解决的自旋系统和保形场理论的方法，实现了这种异构粒细胞系统的扩展。第三，他阐明了tau功能和Whitham变形在四维超对称和方向尺度理论中的作用。最后，他考虑了椭圆形的Calogero-Moser系统，该系统也与四个维度超对称仪的理论密切相关，并发现这些系统的非自治类似物为圆环上的异构层状真皮提供了一个示例。先验数理论与非线性动力学系统，Celular Automata，智能代理，低纤维化和双曲线方程有关，具有Infnite退化性，